Design approach of components using formulas of ...

Design approach of components using formulas of Residual strength function Abir Ghosh

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Abstract:Generally, design of welded components comprises of a static and a fatigue analyses, with one of these, governing design. Residual strength function gives the variation of strength during the intermediate life. The residual strength curve can be used to design more effectively. In this paper, the way design can be carried out based on equations of residual strength previously determined is discussed. For iterations, the Newton-Raphson method is used.

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1. Introduction In normal structural design of components, a static and a fatigue analysis are carried out separately. Fatigue design normally involves the use of S-N curves. Among the two analysis the more critical one governs design. Using fracture mechanics normally involve considering the maximum crack size that NDT can miss as the initial crack length. The known number of cycles are then used to find the crack size to which the crack would propagate. Then the strength at this crack length is determined and compared to the maximum overload. If the strength is higher the component is deemed safe. This approach of using fracture mechanics is more about assessing rather than design. If cracks are found when components are in use then strength and remaining life is determined. Fatigue evaluation started with S-N curves. After initial studies the effect of mean stress was studied. Also strain based curves were investigated. However, it became apparent that internal processes leading to failure need to be understood. Hence crack initiation and propagation became a subject of interest. Meanwhile studies on fracture of a cracked component had been undergoing resulting in fracture mechanics. Crack propagation was understood based on stress intensity, a parameter used in fracture studies. All these fields have developed substantially since they were introduced. The studies in fracture and fatigue crack propagation are distinct. Although the stress intensity concept is used in crack propagation, fracture and crack propagation are largely separate studies. The author has earlier proposed that combining fracture with crack propagation at all phases of the fatigue life will give the residual strength variation during the fatigue life. Apart from unstable crack propagation components can also fail by plastic deformation. Hence residual strength has to be found for both failure modes. The actual residual strength is found by juxta positioning the residual strength of both modes. The residual strength function has been obtained for some common components in fracture. For components failing by unstable crack propagation, residual strength can be differentiated which 3

helps determining inspection periods. Apart from these also other studies on both failure modes have been carried out. In this paper, a procedure of finding design thickness based on equations of residual strength function earlier determined is discussed. The discussion is limited to unstable crack propagation failure only. It is considered that the initial crack length is the maximum crack length that NDI can miss. The number of cycles of application of constant loading is considered to be known along with the design overload. Alternately, for some type of components, the initial crack length can be obtained from S-N curves. For these types of components as well, the design thickness can be determined from the known number of cycles and design overload. It is to be noted that the design method uses a design overload instead to a maximum overload. A description of the design overload is given.

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Design Approach

It needs to be mentioned that here we are designing for a design overload and given number of cycles, which may not be the maximum overload as shown in Fig.1. Residual strength function can allow use of higher load than the design load at intermediate periods.

Fig.1 Designing using residual strength

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The extent to which higher loads can be applied is given by a variable relation between overload and number of cycles. The design overload has to be chosen based on the highest overload near the end of the fatigue life. It need not be the absolute maximum as shown in Fig.1. The method allows the advantage of knowing the residual strength to be utilized. Unnecessary larger design sections can be avoided in this way. However, with introspection one can see that the procedure will only be of advantage if the higher overloads occur early in life. The design overload can be chosen carefully. A proper choice would mean that higher loads applied earlier in life remain below the curve of overload against number of cycles. If that is not the case then a higher design overload need to be chosen. It is also assumed that the overloads are small in number and constant amplitude crack propagation can give the residual strength variation appropriately.

3. Residual strength equations for design The residual strength equations determined earlier need to be modified for use in design. Some modified equations are given in this section. The equations of residual strength function determined earlier give the variation of the number of cycles with strength. For design purpose, we are more interested in the reduction of strength with number of cycles and determining appropriate dimensions so that the reduced strength can sustain the applied overloads.

3.1 Equations for constant magnification factor ‘M’ in stress intensity equation The equations given below is for components failing by unstable crack propagation with stress intensity given by

. The magnification factor M is considered to be constant. This is the

case of a centre or edge crack in an infinite plate, a through crack under internal pressure (infinite case) and an elliptical crack with fixed a/c ratio. The relation for Paris’, Walker’s and Elber’s equation is given in turn.

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3.1.1 Relation for Paris’ Equation

Fig.2 Notation used for design

The above diagram shows that we need to design based on the residual strength variation. Ps is the initial strength. It reduces to the strength Pe after application of Ne cycles of constant amplitude loading. Na has been termed the asymptotic endurance, which would be endurance if the crack propagated through the entire thickness. The equation of residual strength for components of constant ‘M’ subjected to fatigue loading was given by

(1) where A and B are constants, Nx is the number of cycles corresponding to the residual strength x. The equation is suitably converted for design purposes by invoking boundary conditions. Using the notations in Fig.2 the equations can be written as

or

(2)

At Pe =0, Ne =-B=Na

(3)

At Ne =0,

(4) 6

Substituting equations (3) and (4) in equation (2) we get or

(5)

The constant -B and hence Na is given as

, c is the constant stress range applied to an edge crack. (6) Substituting we get

(7)

Also

(8)

Substituting, we get

(9) Ps and Pe are strength (in terms of stress) and c is the constant stress range applied as earlier mentioned. Writing the stresses in terms of shear flow we can write the above equation as

(10)

Or

(11)

Where Ps t=Qs, Pe t=Qe, tc=Qc, t being the thickness for which design is being carried out. Eqn.10 and 11 consider the fracture toughness to be constant which is not the case for a component with 7

thickness less than plane strain value. Either a lower value of K c corresponding to the thickness (such as plane strain value) can be adopted or for better prediction a formula giving a relation between K c and thickness can be inserted into Eqn.11. It is to be noted that not only Kc is affected but Ps in Eqn.11 also changes. This also applies to all the design equations given in this paper. An equation for variation with thickness was given by Irwin as

correction based on a relationship proposed by Vroman is given as

where

.

The thickness

=

,

is the plane strain thickness and Bk and Ak are constants. For elliptical cracks

Ke can be used. If m=3, the above equation becomes

or

(12a)

If thickness is greater than plane strain. For thicknesses, less than plain strain using Irwin’s formula we get,

(12b)

Where Ps1 is the initial strength in terms of stress if the plane strain fracture toughness is taken as fracture toughness. If m=4, the above equation becomes

, or

(13a)

for thicknesses, greater than plane strain

and

(13b)

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for thicknesses less than plane strain, with Ps1 denoting the same parameter discussed earlier. In all the above 3 cases the required thickness can be obtained. Qe is the shear flow corresponding to the design overload. All other parameters are also considered known. Once the thickness is obtained Eqn.10 can be written in the following form

(14)

Here Q’e and N’e are considered as variables. N’e is any number of cycles less than N e, the design endurance and Q’e is the corresponding overload (in terms of shear flow) that can be applied. Hence the overload need not be limited to the design overload (in terms of shear flow) Q e. As long as the overloads applied is within the curve given by Q’ e and N’e the strength of the component is higher. In a similar way Eqn. 11 and 12 can be written with Q’e and N’e considered as variables. In this manner knowledge of residual strength helps with better design. It is to be noted that all the above formulas are given for an edge crack but can be easily modified for a central crack. 3.1.2 Relation for Walker’s Equation The crack propagation equation due to Walker is given as 2a

2a x

x da da p  K   Nx   N  1  R , where or K  x n  p max C (K ) p  n 1 R 2 ai CKmax K  2 ai

(15)

If Walker’s is used as the crack propagation equation, then the residual strength relation can be written as

(16)

where

Na 

 pn   1 2 

ai 

1  R  p

  p  n  pn 1   2  C      

pn 2

for an edge crack. The design equation is given as

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or

(17)

3.1.3 Relation for Elber’s Equation The crack propagation due to Elber which takes into account crack closure is given as Elber’s equation can be written as

Nx 

2ax

da  C K  , K eff  UK n

2 ai

where U  0.5  0.4R as given by Elber or values proposed

eff

by other researchers

(18)

If Elber’s equation is used as the crack propagation equation, then the residual strength relation can be written as

(19) where N  a

a i 1n / 2  n C U     n / 2 n / 2   1 n

The design equation is given as

or

(20)

Again, as in the Paris’ eqn. the formulas considering Walker’s Eqn. and Elber’s Eqn. is given for an edge crack but can easily converted to that for a central crack.

3.2 Equations for the strip yield model

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Equations of the strip yield model are amenable to the design format discussed here. For the strip yield model, residual strength function is given as

(21)

Where Aa and Ba are constants. Using the notations used in Fig.2 the above equation can be written as

(22)

Using the boundary conditions as earlier we get

At

(23)

(24)

Substituting Eqns. 22 and 23 in 21 we get

(25)

Where 1 and 2 are the higher and lower values of the stress range. The crack considered is a centre crack. Substituting the above value of Na and using Eqn.8 we get

(26)

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Writing the above equation in terms of shear flow we get

(27)

Where Q1 and Q2 are the shear flows corresponding to the stresses 1 and 2.

4. Initial crack length and thickness

Consider the case of designing based on NDT. In this case the maximum crack length that NDT could miss can be considered to be the initial crack length. Hence ai is known and the initial strength of a component to be designed in terms of stress is known. The initial crack length can be for some components found from S-N curves. For example, in the case of steel components for some classes in codes such as BS 5400 the endurance is given as proportional to the cube of stress range. If Paris equation is used then the exponent in Paris’ equation is also 3 for common steel. If both the crack propagation equation and the endurance is dependent on the same exponent then S-N curves can be used to determine a constant value of initial crack length. In such cases if the endurance is dependent on thickness then the initial crack length also would vary with thickness. Thus, Qs would be given as a function of thickness. Hence there are three cases mentioned above, the first involving NDT and the next two involving S-N curves. To solve for the thickness ‘t’ one needs to use an iterative procedure. A technique such as the Newton-Raphson’s method can be used.

5. Illustrations Consider a plate of 15-5PH H900 steel which has a Paris exponent of 4. The fracture toughness is 1709.05 N/mm3/2 and the yield strength is 1168.65 N/mm2 (as per NASGRO tables). The constant C in Paris equation is 1.45E-15 in SI units. The design overload is considered to be 4000 N/mm. The component is considered to be subjected to an amplitude loading of 2000 N/mm for 10 4 cycles. Also 12

overloads of 6000 N/mm and 5500 N/mm occur after 102 and 103 cycles. The initial crack length is considered to be an edge crack of 2.5 mm. R is taken as 0.4. The threshold is determined as 115 N/mm3/2. Failure of the component is governed by unstable crack propagation and the plate is infinite. The initial strength of the component is given by

. Here M for an edge crack is taken

as 1.12. Hence Ps is obtained as 544.457 N/mm2. The plane strain thickness is 5.35 mm.

Using

Eqn.13a and substituting the respective values we get the equation in ‘t’ of 2.96E+05t41.6E+07t2=3.40E+09. The equation can also be written considering t as variable as Ps2t4-Qet2constant=0. Considering this equation as f(x), we get f’(x)=4Ps 2t3-2Qet=0. We start with a thickness ‘t’ of 14 mm and after four iterations using Newton-Raphson’s method the thickness is obtained as 11.72 mm. The initial stress intensity is found to be high enough compared to the threshold for the crack to propagate. Since the thickness calculated is higher than the plane strain thickness, the problem does not have the additional issue of a varying toughness. Eqn.14 can be used to check for the two overloads mentioned at 10 2 and 103 cycles. Eqn.14 is obtained as 4.07E+07-Q’e 2=2.47E+03N’e. At N’e of 103 cycles the maximum overload is determined as 6184 N/mm and at 102 cycles the maximum overload is determined as 6361 N/mm. Hence the calculated thickness would be able to sustain the overloads mentioned. Next consider a component of A 542 Cl3 steel with an edge crack. The plane strain fracture toughness is 2050.86 N/mm3/2. The Paris exponent is 3 and the constant C in SI units is 5.088E-13. The yield is 515.58 N/mm2. The maximum overload is taken as 4000 N/mm and the range of load as 2000 N/mm. Again, an edge crack of 2.5 mm is taken and the component is considered to be infinite and fail by unstable crack propagation. R is taken as 0.4. The threshold is determined as 94 N/mm3/2. The number of cycles applied is taken as 104. The plane strain thickness is determined as 39.55 mm. It is considered that the design thickness might be less and we need to deal with varying fracture toughness. Hence Eqn.12b is used. The 13

value of Ps1 is calculated as 653.35 N/mm2. We consider Eqn.12b as f(x). The equation is then differentiated to obtain f’(x). Eqn.12b attains the form 4000t3-653.35t3xSQRT(t2+350.49)-1.17E+06x SQRT(t2+350.49)=0. Starting with a value of t=12mm we substitute it in f(x) and f’(x) to find the value of t for the next iteration using the Newton-Raphson method. After four iterations design thickness is obtained as 13.46 mm. The initial stress range is found higher than the threshold and hence the crack would propagate. Consider a third case of a plate with a central crack of 2.5 mm. The problem is solved using the strip yield method. The plate is considered to be of 15-5PH H900 steel which was also used in the first illustration. The loading range is considered to vary from 500 N/mm to 2500 N/mm and the maximum overload is considered to be 5000 N/mm. The threshold is 116 N/mm3/2. The component fails by unstable crack propagation and is considered to be subjected to 10 4 cycles. The initial strength is calculated as 862.376 N/mm 2. Since the plain strain thickness is comparatively small, the design thickness was estimated to be in the plain strain range. The value of m=4 is substituted in Eqn.27 and the design thickness is found to be 9.73 mm. The range of stress intensity is higher than threshold and the crack propagates.

6. Conclusions Design of components for fatigue is generally carried out using S-N curves. Either the S-N curve design or static design governs. Often components are assessed using fracture mechanics. If NDT is used then the largest crack that NDT would miss is considered as the initial crack length and the life determined. Also, if cracks are identified then the remaining life is determined in a similar manner. Residual strength function involves combining fracture with crack propagation to give the variation of strength during the entire life of a component. Residual strength functions have been obtained for some common cases and crack propagation laws. The use of some of these equations for carrying out design is discussed in this paper. We might consider the initial crack to be a fixed value

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independent of thickness as in the case of determining life based on NDT. Alternatively, for some cases, initial crack lengths may be obtained from S-N curves. Since residual strength variation was not available before, the variation of strength could not be used for design. The use of residual strength curves allows us to design for minimum dimensions. If larger overloads occur early in the fatigue life then we can design for a lesser overload later in the fatigue life and later check for the larger overloads. Here, some equations have been suitably altered to be able to be used for design. The modification is carried out by substituting the values of strength when endurance is zero and the supposed endurance when strength is zero. The equations are then written in terms of a design thickness. The initial case taken is that of components which can be given by the equation

where

M is constant. This includes a centre crack or edge crack in an infinite plate, through crack under internal pressure (infinite case) or an elliptical crack with constant a/c ratio. The crack propagations equations considered are Paris Eqn., Walker’s Eqn. and the equation including crack closure due to Elber. Next, an equation that can be used for design for the strip yield method is given. Paris’ equation is considered for crack propagation. Three illustrations are given for materials with m=4 and 3 in Paris’ Eqn. The first case shows the way that higher overloads occurring earlier in life can be checked. The second case deals with plane stress and a varying fracture toughness. The third case deals with the strip-yield method. The Newton Raphson method is used for iterations.

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References

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 

nn / 2  1 4 2 5 n / 2 Paris, P. C., Gomez, R. E., and Anderson, W. E. (1961). “A rational analytic theory of 2 162b  fatigue,” The Trend in Engineering, University of Washington, 13, pp 9-14.

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